Risk and return
Table of Contents
A. Risk and Return: Past and Prologue
I. Fundamentals Review
1. Expectation
In the scope of investments, expectation is the investors' forecast of different states of the economy in the future.
 Ex ante: Before the event.
 Measures the expected return before the event.
 Ex post: After the event.
 Measures the expected return after the event.
State of the economy  Probability  Return 

Recession  20%  5% 
Normal  50%  5% 
Expansion  30%  15% 
$$\textbf{E(r)} = \Sigma_{s} \ p(s) \times [r_s]$$
Variance is an important measure of uncertainty. Gives a quantitative measure of how much realized deviates from what expected. This is also known as volatality.
$$\sigma^2 = \Sigma_{s} \ p(s) \times [r_s  \text{E(r)}]^2$$
2. Expost average returns & $\sigma$
Expost: After the fact, at t (today):
 Use the past to predict the future.
 Empirically, we use ex post (history) average returns to approximate ex ante expected returns.
$$\bar{r} = \Sigma^T_{t=1}\frac{\text{HPR}_t}{T}$$
Where, $T$ is number of observations and $\bar{r}$ is the average HPR.
Expost variance: $$\sigma^2 = \frac{1}{T1} \Sigma_{i=1}^T (r_i\bar{r})^2$$
Expost standard deviation: $$\sigma = \sqrt{\sigma^2}$$
Annualizing the statistics:
$$\bar{r}_{\text{annual}} = \bar{r}_p \times \text{periods} $$
$$\sigma_{annual} = \sigma_{period} \times \sqrt{\text{# periods}}$$
3. Normal distribution
In finance, we use the normal distribution to approximate the possible outcomes investors face.
 Risk is the difference between the expected return and the realized (actual) returns.
Normal distributions have key characteristics that are easy to spot in graphs:
 The
mean
,median
andmode
are exactly the same.  The distribution is symmetric about the mean—half the values fall below the mean and half above the mean.
 The distribution can be described by two values: the mean and the standard deviation.
Empirical rule
The empirical rule, or the 689599.7 rule, tells you where most of your values lie in a normal distribution:
 Around 68.26% of values are within 1 standard deviation from the mean.
 Around 95.44% of values are within 2 standard deviations from the mean.
 Around 99.74% of values are within 3 standard deviations from the mean.
 $E+(x)\sigma$ are the best cases
 $E(x)\sigma$ are the worst cases
Hence, the attractiveness of an investment depends not just on the expected returns, but also the risk of losses.
II. Arithmetic vs. Geometric
When evaluating past performance, the geometric average is more precise in describing performance when volatility is high.
When statistically calculating expected returns using historical data, use simple average (Arithmetic) often because it is convenient, and it has nice statistical properties.
 Large sample theory: Sample size n grows indefinitely.
 Law of large numbers tells that the sample mean of a random variable converges to its true mean for a large sample.
III. Exante Risk Premium & Risk Aversion

Risk premium: The difference in expected return between risky and riskfree investments.

How much I would like to have at stake to be able to get high returns

Why do invester require it?
 Investors are risk averse hence require a compensation to take on risk.
 (aka) average excess return


Riskfree rate $(r_f)$: Rate of return that can be earned with certainty.
B. Inflation and Real Rates of Return
Taxes are paid on nominal investment income. This reduces real investment income even further.
 nominal rate of return is amount of money generated by an investment before factoring in expenses such as taxes and inflation.
$$r_{\text{real}} \approx [r_{\text{nominal}}\times (1\text{tax %})]  \text{Inflation rate}$$
C. Asset Allocation Across Risky and RiskFree Portfolios
How much of your wealth should you invest in risky assets?
The amount you invest would depend on:
 Risk
 Risk premium  how much do investors get compensated for risk
 Risk aversion  How much do investors dislike risk?
I. Complete Portfolio
The entire portfolio including risky portfolio (RP) and riskfree portfolio (RF).
$$E(r_{\text{complete}}) = yE(r_p) + (1y)r_f$$
Riskfree asset, $r_f$: proxy with TBills/money market instruments.
$y$ is called the weight in the risky portfolio.
$$\sigma^2_{\text{complete}}=(y\sigma_{rp})^2 + [(1y)\sigma_{\text{rf}}]^2$$
$\sigma_{\text{rf}}=0$ as the return from a riskfree asset is certain.III. Capital Allocation Line (CAL)
$$E(r_c) = r_f + \frac{\sigma_c}{\sigma_p}\times(E(r_p)r_f)$$
The slope of the CAL measures the tradeoff between risk ($\sigma_p$) and risk premium ($E[r_p]r_f$).
The slope is known as the Sharpe ratio.
1. Using Leverage with CAL
Assumption: Borrow at riskfre rate. (To ensure that the line is linear)
Borrow at the riskfree rate and invest in risky portfolio. With 50% leverage: $y=1.5$
Own money = $10,000
Borrowed money = $5,000
$$y=\frac{\text{own money}+\text{borrowed money}}{\text{own money}}$$
The line should be straight because for the following simple reasons.
 It could be kinked line if borrowing cost is higher than Rf.
IV. Risk Aversion and Allocation
Risk aversion is the willingness to trade off risk against expected return.
 Individual investors have different levels of risk aversion.
V. Quantifying Risk Aversion
Meanvariance analysis is one part of modern portfolio theory.
Assumption: Investors seek low risk and high reward.
 Variance may tell how spread out the returns of a specific security are on a daily or weekly basis.
 Expected return is a probability expressing the estimated return of the investment in the security.
$$A = \frac{E(r_c)r_f}{\sigma_c^2}\gt0 \ \text{if risk adverse}$$
 Large A means that investor require additional return to bear risk
Average risk aversion = 1.5 to 4 (Empericallly)
A = 0; Risk neutral A < 0; Risk loving
$$y_{\text{percent in risky asset}} = \frac{(E(r_p)r_f)/\sigma_p^2}{A_i}$$
D. Efficient Diversification
I. Asset Allocation With Two Risky Assets
1. Expected Return: Two Risky Assets
$$E(r_p) = W_1E_1(r_1)+W_2E(r_2)$$
2. Portfolio Variance and Standard Deviation: Two Risky Assets
$$\sigma^2_p = W_1^2\sigma^2_1 + W_2^2\sigma^2_2 + 2W_1W_2\text{Cov}(r_1,r_2)$$
$$Cov(R_A, R_B) = \frac{\Sigma_i^n(R_{A_i}\bar{R_{A}})\times(R_{B_i}\bar{R_{B}})}{n1}$$
3. Relation between Covariance & Correlation Coefficient
 Correlation coefficient $\rho$
$$\rho_{1,2} = \frac{\text{Cov(r_1, r_2)}}{\sigma_1 \times \sigma_2}$$

Correlation measures the intensity and direction that two stocks tend to move together

Correlation is standardized to be between 1 and 1
4. Correlation Coefficient ($\rho_{1,2}$)

If $\rho_{1,2}=1.0$  Perfectly positively correlated
 $\sigma(1,2) = W_1\sigma_1 + W_2\sigma_2$
Correlation is standardized to be between 1 and +1.
 If $\rho_{1,2}=1.0$  Perfectly negatively correlated
 $\sigma(1,2) = \pm(W_1\sigma_1  W_2\sigma_2$)
$$\sigma^2_p = W_1^2\sigma^2_1 + W_2^2\sigma^2_2 + 2W_1W_2\text{Cov}(r_1,r_2)$$
$$\text{Cov}(r_1, r_2)=\rho_{1,2}\sigma_1 \sigma_2$$
$$\sigma^2_p = W_1^2\sigma^2_1 + W_2^2\sigma^2_2 + 2W_1W_2\rho_{1,2}\sigma_1 \sigma_2$$